/** * Verify the input arguments are valid for correct use of the bisection * method. If the tolerance is less than zero, an Exception will be thrown. * If f($a) and f($b) have the same sign, we cannot use the intermediate * value theorem to guarantee a root is between $a and $b. This exposes the * risk of an endless loop, so we throw an Exception. If $a = $b, then clearly * we cannot run our loop as $a and $b will themselves be the midpoint, so we * throw an Exception. * * @param Callable $function f(x) callback function * @param number $a The start of the interval which contains a root * @param number $b The end of the interval which contains a root * @param number $tol Tolerance; How close to the actual solution we would like. * * @throws Exception if $tol (the tolerance) is negative * @throws Exception if $a = $b * @throws Exception if f($a) and f($b) share the same sign */ private static function validate(callable $function, $a, $b, $tol) { Validation::tolerance($tol); Validation::interval($a, $b); $f⟮a⟯ = $function($a); $f⟮b⟯ = $function($b); if (Special::sgn($f⟮a⟯) === Special::sgn($f⟮b⟯)) { throw new Exception\BadDataException('Input function has the same sign at the start and end of the interval. Choose start and end points such that the function evaluated at those points has a different sign (one positive, one negative).'); } }
/** * Verify the input arguments are valid for correct use of fixed point * iteration. If the tolerance is less than zero, an Exception will be thrown. * If $a = $b, then clearly we cannot run our loop as [$a, $b] will not be * an interval, so we throw an Exception. If $a > $b, we simply reverse them * as if the user input $b = $a and $a = $b so the new $a < $b. * * @param number $a The start of the interval which contains a root * @param number $b The end of the interval which contains a root * @param number $p The initial guess of our root * @param number $tol Tolerance; How close to the actual solution we would like. * * @throws Exception if $tol (the tolerance) is negative * @throws Exception if $a = $b * @throws Exception if either $p > $a or $p < $b return false */ private static function validate($a, $b, $p, $tol) { Validation::tolerance($tol); Validation::interval($a, $b); if ($a > $b) { list($a, $b) = [$b, $a]; } if ($p < $a || $p > $b) { throw new Exception\OutOfBoundsException("Initial guess {$p} must be in [{$a}, {$b}]."); } }